Simplify; express your answer in exponential form. Assume $q\neq 0, p\neq 0$. $\dfrac{{(q^{-4}p^{-5})^{-5}}}{{(q^{-1}p^{-4})^{5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{-4}p^{-5})^{-5} = (q^{-4})^{-5}(p^{-5})^{-5}}$ On the left, we have ${q^{-4}}$ to the exponent ${-5}$ . Now ${-4 \times -5 = 20}$ , so ${(q^{-4})^{-5} = q^{20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{-4}p^{-5})^{-5}}}{{(q^{-1}p^{-4})^{5}}} = \dfrac{{q^{20}p^{25}}}{{q^{-5}p^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{20}p^{25}}}{{q^{-5}p^{-20}}} = \dfrac{{q^{20}}}{{q^{-5}}} \cdot \dfrac{{p^{25}}}{{p^{-20}}} = q^{{20} - {(-5)}} \cdot p^{{25} - {(-20)}} = q^{25}p^{45}$